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Compressive sensing of binary signals is corresponding to the problem of binary symbol detection in the faster-than-Nyquist signaling systems, which has significant research value. Traditional compressive measurement of a binary signal is based on Gaussian matrix, and l1 minimization is a classic algorithm for signal reconstruction. However, stochastic matrix such as the Gaussian matrix can hardly be realized by a digital circuit, and the reconstruction performance of l1 minimization is not well enough for binary signals. Thus, it is of great meaning to construct a new kind of measurement matrix as well as a better reconstruction algorithm for binary signals. This paper constructs a chaotic circulant measurement matrix based on Cat chaotic map (CCMM), and proposes a brand new algorithm for binary signal reconstructionsmooth function approximation method (SFAM). Chaotic sequence has characteristics of both internal certainty and external randomness, while a circulant matrix requires less elements and can be realized through fast Fourier transform. CCMM conbines the advantages of both chaotic sequence and circulant matrix, so that it not only satisfies the RIPless property required by the compressive measurement matrix because of external randomness, but also has the power to resist the effect of low signaling efficiency and low SNR due to the internal certainty. Moreover, the circle structure gives CCMM the potential to be digital realized in the future. In SFAM, we first use a non-convex function to approximate the original discontinuous objective function, in order to transfer the original combinatorial optimization problem into an optimization problem with equality constraints which can be solved much easier. Then we use the interior point method to solve this optimization problem. Furthermore, sparse Bayesian learning algorithm is used to correct the reconstruction error for a more accurate result. Compressive measurement and reconstruction of binary signals in additive Gaussian white noise channel are operated. Result of numerical experiments shows that CCMM is much better than the traditional Gaussian matrix for compressive measurement, especially in the condition of low signaling efficiency and low SNR, and SFAM is much better than l1 minimization for binary signal reconstruction. At the end of this paper, we explain the essential reason why CCMM performs better than the traditional Gaussian matrix, through calculating the autocorrelation function of compressive measurement vector in various conditions.
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Keywords:
- compressive sensing /
- chaotic circulant matrix /
- smooth function approximation method /
- binary signal
[1] Donoho D L 2006 IEEE Trans. Inform. Theory 52 1289
[2] Cands E J 2008 Comptes. Rendus Math. 346 589
[3] Cands E J, Wakin M B 2008 IEEE Sig. Proc. Mag. 25 21
[4] Lustig M, Donoho D L, Pauly J M 2007 Magn Reson Med. 58 1182
[5] Sen P, Darabi S 2011 IEEE Trans. Vis. Comput. Graphics. 17 487
[6] Hrman M A, Strohmer T 2009 IEEE Trans. Signal Process. 57 2275
[7] Hajela D 1990 IEEE Trans. Inf. Theory 36 289
[8] Rusek F, Anderson J B 2009 IEEE Trans. Commun. 57 1329
[9] Sugiura S 2013 IEEE Wireless Commun. Lett. 2 555
[10] Yin W, Morgan S, Yang J F, Zhang Y 2010 Rice University CAAM Technical Report TR10-01
[11] Yin W, Osher S, Xu Y Y 2012 Inverse Probl. Imag. 8 901
[12] Cands E J, Plan Y 2011 IEEE Trans. Inform. Theory 57 7235
[13] Guo J B, Wang R 2014 Acta Phys. Sin. 63 198402(in Chinese) [郭静波, 汪韧 2014 63 198402]
[14] Guo J B, Xu X Z, Shi Q H, Hu T H 2013 Acta Phys. Sin. 62 110508(in Chinese) [郭静波, 徐新智, 史启航, 胡铁华 2013 62 110508]
[15] Allen Y. Y, Zhou Z H, Arvind G B, S. Shankar S, Ma Y 2013 IEEE Trans. Image Process. 22 3234
[16] Chen S S, Donoho D L, Saunders M A 1998 SIAM J Sci. Comput. 43 33
[17] Lu C W, Liu X J, Fang G Y 2011 Acta Electronica Sinica 39 2204 (in Chinese) [卢策吾, 刘小军, 方广有 2011 电子学报 39 2204]
[18] Chen G, Mao Y, Chui C K 2004 Chaos Soliton Fract. 21 749
[19] Mohimani H, Babaie-Zadeh M, Jutten C 2009 IEEE Trans. Signal Process. 57 289
[20] Wipf D P, Rao B D 2004 IEEE Trans. Signal Process. 52 2153
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[1] Donoho D L 2006 IEEE Trans. Inform. Theory 52 1289
[2] Cands E J 2008 Comptes. Rendus Math. 346 589
[3] Cands E J, Wakin M B 2008 IEEE Sig. Proc. Mag. 25 21
[4] Lustig M, Donoho D L, Pauly J M 2007 Magn Reson Med. 58 1182
[5] Sen P, Darabi S 2011 IEEE Trans. Vis. Comput. Graphics. 17 487
[6] Hrman M A, Strohmer T 2009 IEEE Trans. Signal Process. 57 2275
[7] Hajela D 1990 IEEE Trans. Inf. Theory 36 289
[8] Rusek F, Anderson J B 2009 IEEE Trans. Commun. 57 1329
[9] Sugiura S 2013 IEEE Wireless Commun. Lett. 2 555
[10] Yin W, Morgan S, Yang J F, Zhang Y 2010 Rice University CAAM Technical Report TR10-01
[11] Yin W, Osher S, Xu Y Y 2012 Inverse Probl. Imag. 8 901
[12] Cands E J, Plan Y 2011 IEEE Trans. Inform. Theory 57 7235
[13] Guo J B, Wang R 2014 Acta Phys. Sin. 63 198402(in Chinese) [郭静波, 汪韧 2014 63 198402]
[14] Guo J B, Xu X Z, Shi Q H, Hu T H 2013 Acta Phys. Sin. 62 110508(in Chinese) [郭静波, 徐新智, 史启航, 胡铁华 2013 62 110508]
[15] Allen Y. Y, Zhou Z H, Arvind G B, S. Shankar S, Ma Y 2013 IEEE Trans. Image Process. 22 3234
[16] Chen S S, Donoho D L, Saunders M A 1998 SIAM J Sci. Comput. 43 33
[17] Lu C W, Liu X J, Fang G Y 2011 Acta Electronica Sinica 39 2204 (in Chinese) [卢策吾, 刘小军, 方广有 2011 电子学报 39 2204]
[18] Chen G, Mao Y, Chui C K 2004 Chaos Soliton Fract. 21 749
[19] Mohimani H, Babaie-Zadeh M, Jutten C 2009 IEEE Trans. Signal Process. 57 289
[20] Wipf D P, Rao B D 2004 IEEE Trans. Signal Process. 52 2153
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